The first couple of times you fight a Watcher Transport, all you have to do is hold out against the waves of enemies it sends at you until the Survivor AA Guns that are surrounding you take it out.

Later, in the escort mission that starts the third Episode of the game, you must destroy a couple of Transports using any means you can before they destroy The Ark. It is preferable to use the AA Guns located on The Ark itself, however you can also use Survivor Airplanes or your own rocket pack to take them out if you wish.